System for diagnosing and treating sleep apnea

ABSTRACT

Two programs measure upper airway resistance using Resistance=Pressure/Flow. Raw flow and pressure data is divided into breaths and time adjusted so each breath starts with the value zero. Each breath is graphed as flow in the y-axis and time in the x-axis. The slope between points Flow=0 and Flow=0.20 is calculated. The resistance is the inverse of the slope. The second program determines whether a breath is flow limited or not. It also uses the flow and time data to perform a curve fitting to describe the flow-time in meaningful polynomial function F (P)=A t 3 +B t 2 +Ct+D. The derivative of this function is F′= 3 At 2   +2 Bt+C. If the value of the derivative F′ at maximum flow is less than or equal zero, then it is a flow limited breath, otherwise, it is non flow limited breath. Also, the need for a pressure-monitoring catheter is obviated.

RELATIONSHIP TO OTHER APPLICATION

This application claims the benefit of the filing date of ProvisionalApplication for U.S. Letters Patent Ser. No. 60/400,038 on Aug. 2, 2002in the name of the inventors herein.

GOVERNMENT RIGHTS

A portion of this invention was made under contract of appointmentawarded by The John D. Dingell Veterans Administration Medical Center inDetroit, Mich. wherein one or more inventors herein stand appointed bythe Veterans Administration (“VA”) to perform work on behalf of the VAin its facilities. As such, the VA appointed inventor(s) are subject tothe federal laws applicable to inventions and discoveries made in thecourse of the VA appointment. The government has certain rights in thecorresponding portion of the present invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to systems for diagnosing sleepingdisorders, and more particularly, to a system that is useful ineffecting rapid diagnosis of, and providing controlled therapy to,patients who suffer from sleep apnea.

2. Description of the Related Art

There is a need in the present state of the art of diagnosing sleepingdisorders for automated non-invasive methods that yield objective andreproducible data responsive to the presence of inspiratory flowlimitation during sleep. There is a particular need for a system thatcan perform the necessary acquisition of such data without requiring theuse of a pressure-monitoring catheter in the pharynx of the patient.

From the anatomical standpoint, the airways consist of upper and lowerairways. Sleep apnea is a common condition that is characterized byobstruction or narrowing of the upper airway. The upper airway segmentsare the nose, the mouth, and the larynx. The larynx opens to the tracheaand branches into two bronchi. Each bronchi enters a lung and terminatesin the alveoli. The analysis that is presented herein in support of thepresent invention focuses on the pharyngeal upper airway. This upperairway consists of the extrthoracic trachea, the larynx, pharynx, andthe nose. The principal site for upper airway closure or narrowingduring sleep is the pharynx, which is a heterogeneous structure, and itis part of the pharyngeal airway.

The pharyngeal airway is divided into four segments. These segments arethe nasopharynx, the velopharynx, the oropharynx, and the hypopharynx.During inspiration, the pharyngeal structure moves forward toward thecenter of the lumen.

In addition, more than twenty skeletal muscles surround the pharyngealairway, the muscles being functionally designated as “dilator” musclesand “constrictor” muscles. The pharyngeal muscles receive aphasicactivation during inspiration and support a patent pharyngeal lumenthrough which air flows. Contraction of the pharyngeal muscles candilate and stiffen the pharyngeal airway, and the constrictor musclescan improve the upper airway patency. The tongue comprises a highlymobile structure that can occlude the pharyngeal airway and the softpalate, which are important in maintaining upper airway patency. It isalso known that the tongue is a major muscle comprising protrude andretractor muscles. Either co-activation of the protrude and theretractor muscles, or independent activation of the protrude muscles,can improve upper airway flow mechanics. Co-activation decreasespharyngeal collapsibility but does not dilate the pharyngeal airway.activation of the tongue protrude muscles results in enlargement of theupper airway.

Before describing the art in greater detail, it may be useful toidentify some of the acronyms that are widely used: Acronym DefinitionAHI → Apnea-hypopnea index ANOVA → Analysis of Variance CHF → ChronicHeart Failure CPAP → Continuous Positive Airway Pressure cRUA →Calculated Upper Airway Resistance DME → Therapeutic Products from aReseller DTC → Direct to Consumer EEG → Electroencephalography EMG →Electromyography EOG → Electrooculography FOT → Forced OscillationImpedance IFL → Inspiratory Flow Limitation IHD → Ischemic Heart DiseaseIPS → DeVilbiss Internet Processing Software MEMS → MicromachinedElectro-mechanical Pressure Sensors mRUA → Measured Upper AirwayResistance NIFL → Non-Flow Limited NPV → Negative Predictive Value NREM→ Stage 2 sleep (non-REM?) OSA → Obstructive Sleep Apnea PAP → PositiveAirway Pressure PAT → Peripheral Arterial Tone PPV → Positive PredictiveValue PSG → Polysomnography REM → Rapid Eye Movement sleep RUA → UpperAirway Resistance SDB → Sleep-Disordered Breathing SPC → SuperiorPharyngeal Constrictor UARS → Upper Airway Resistance Syndrome

Studies of OSA subjects indicate that activation of the superiorpharyngeal constrictor (“SPC”) muscle is similar to the action ofpharyngeal dilator muscles during spontaneous and induced apneas.However the effect of each single muscle in regard of sleep obstructionis not yet clear. It has been reported that the mechanical properties ofthe upper airway is independent of the dilator skeletal muscles thatsurround it. Also, the prior art is asserted to have determined thatpressure is correspondingly equivalent to volume expansion. In otherwords, although the specific effect of each muscle is not clear in thepresent state of the art, the ratio of pressure to volume expansion is1:1, irrespective of whether the dilator muscles are in active orpassive conditions.

Inspiratory flow limitation (“IFL”) is the mechanical corollary ofsnoring and corresponds to a narrowing of the upper airway of a patient.The detection of inspiratory flow limitation will improve the diagnosisof sleep disordered breathing. In the present state of the art, thedetection of inspiratory flow limitation typically is achieved by atrained observer who analyzes each breath individually. Such a trainedobserver will study visually the shape of the time-flow curve thatcharacterizes each breath under consideration. This labor intensivemethodology, particularly since it applies a measure of subjectivity toa relatively few breaths, is not adequately precise to achieve anobjective and fully reproducible determination of the presence of airflow limitation, and will often result in false diagnoses.

There are available in the art standardized procedures for detectingflow limitation based on analysis of the pressure-flow aspect of therespiration cycle. The known methods, however, are practicable only insleep research laboratories, and typically are not available in theclinical sleep lab. In the clinical environment, a physician typicallywill spend about two hours to analyze only about 40 patient breaths.Clearly, the known process is time consuming, functions on a verylimited data set, and is likely to produce erroneous results.

Obstructive steep apnea is a condition that is characterized bycessation of breathing during sleep, the air flow being obstructed inthe upper airway of the subject. IFL during sleep is defined asdecreasing supraglottic pressure without corresponding increase inairway flow rate. This condition generally causes repetitivedisturbances during sleep resulting from inadequate flow of air into thelungs of the subject. It is known in the art that resistance to the flowof air is increased during the transition from a state of wakefulness tosleep. A characteristic cross-sectional area can readily be determinedin relation to the linear portion of the pressure-flow loop, i.e.,corrsponding to a progressive increase in air flow resistance.

There is present in this model a coupling condition that is responsiveto the characteristics of the solid structure (i.e., tissue and muscles)of the upper airway and the air flow. The solid structure ischaracterized by the arrangement of the muscles, the tissue structure,and the viscous flow, which is the air flow. The state of the art issuch that there is no indication that muscles affect the compliance ofupper airway. The foregoing notwithstanding, the entire solid tissuestructure might have a significant effect on obstruction depending uponthe viscoelastic properties. However, since such viscoelastic propertiesof the upper airway have not been thoroughly studied in the relevantliterature, it is expected that an analysis based on such propertieswould have unacceptable uncertainties in its results. Accordingly,greater certainty is achieved if the analysis approaches the problemfrom the standpoint of air flow in a collapsible tube.

Upper airway obstruction can be caused by several factors. For example,studies have shown that patient with obstructive sleep apnea have anupper airway cross sectional area that is less than that of normalsubjects. Therefore, subjects with small upper airway cross sectionalarea are more likely to have sleep obstruction.

Adipose tissue is the connective tissue in which fat is stored. Thistissue surrounds the pharyngeal upper airway, and it has been assertedthat this fat might, due to gravity and mass loading that act on thelumen, reduce the upper airway cross sectional area. Other studies,however, assert afinding that the deposit of fact is not related to thepharyngeal narrowing. Instead, the narrowing depends on the thickness ofthe muscles. Yet another study shows that there are more fat depositsaround the collapsible pharyngeal upper airway in patients diagnosedwith obstructive sleep apnea, compared to normal subjects. The crosssectional area of patients with Obstructive Sleep Apnea (OSA) is alsoless than normal during wakefulness. The foregoing notwithstanding, themechanical effect resulting from fats deposits has not yet beendetermined.

Mucosal adhesive forces have been considered in the art. The wall of theupper airway has a lining of mucus. It has been hypothesized thatsurface adhesive forces plays an important role in determining themechanical properties of the upper airway. During narrowing this mucuslining add more thickness to the surface of the wall, andcorrespondingly, surface adhesive forces increase the pressure. Surfaceadhesive forces are considered important in determining the magnitude ofthe opening pressure required to prevent the mucus from maintainingcontact. In animal experiments, it was found that adding topicallubricant to the upper airway reduce the severity It has also beensuggested that decreasing the adhesive characteristic of the surfaceforces will render the upper airway more resistive to collapsing. Thus,it has been hypothesized that the secretion of the mucus could be animportant factor during closure of the upper airway, because this willdetermine the magnitude of the air pressure required to reopen theairway. In sum, however, the physics in support of reduction of theadhesive forces has yet to be explained.

Others in the art have asserted that pressure gradient (i.e., transmuralpressure), plays a role in in the collapsibility of the upper airway.Transmural pressure is the pressure difference between intraluminalpressure (pressure inside the lumen), and the extraluminal pressure(atmospheric pressure). During inspiration the pressure in thepharyngeal upper airway is reduced as the cross sectional area isdecreased, and the velocity of the air is correspondingly increased. Asthe pressure in the thorax region increases negatively in relation toatmospheric pressure, the velocity of the air is reduced. Thepressure-velocity relationship is consistent with the well-knownBernoulli equation. Thus, a reduction in the transmural pressure willresult in a decreased cross-sectional area. The transmural pressure isdefined as intraluminal pressure (P_(t)) minus surrounding tissuepressure (P_(t)). Cross sectional area can be minimized as thetransmural pressure decreases. The prior art has speculated that thereduction or occlusion in the luminal cross-section is a result ofnegative intraluminal pressure, or positive (P_(t)). Therefore, astransmural pressure (P_(tm)) increases, the cross sectional areaincreases, and vice versa. During narrowing the intraluminal pressure isalways negative, and tissue pressure is positive (with respect toatmospheric pressure).

Thoracic caudal traction is caused by inspiratory thoracic activityresulting from an increase in the cross-section of the upper airway.Caudal traction has two mechanisms. The first mechanism is characterizedby the stiffening of the upper airway as longitudinal tension is appliedto the upper airway. The second mechanism is the dilation of thepharyngeal airway. It has been reported in the art that caudal tractionmight transmit sub-atmospheric pressure to the tissue that surrounds theupper airway. In this situation the transmutable pressure increases,whereby the difference between intraluminal pressure, and tissuepressure increases radially, (P_(tm)=P_(interlunimal)−P_(tissue)) andthe collapsibility of the upper airway decreases. Others have found anincrease in the maximum inspiratory flow upon observing a decrease inthe collapsibility of the upper airway with tracheal displacement.Caudal traction has been shown to increase the maximum V_(I Max)inspiratory flow at all levels of tongue displacement. V_(I Max) wasnoticeably increased for each tongue and caudal displacementinteraction. Caudal displacement affects the critical pressure responseof the tongue displacement. Thus, under caudal traction there isobserved a corresponding decrease in the critical value, while thetongue was displaced.

Upper airway resistance is an important mechanical characteristic forthe airflow in the upper airway. Resistance has been described as anindirect measure for upper airway caliber, and has been asserted to bethe most significant factor in determining the upper airway caliber, asmeasured on the linear portion of the pressure-flow curve. However, thedetermination of resistance is not valid if flow limitation is presentbecause pressure and flow are not associated any more.

It has been reported that the resistance of the upper airway increasemore than normal prior to obstruction. Upper airway resistance syndromeis characterized by repetitive episodes of IFL, and decreases inesophageal pressure leading to recurrent arousal. High upper airwayresistance can cause tiredness, excessive daytime sleepiness, and achange in blood pressure. Subjects with high resistance work harder atbreathing. It also has been reported that a significant increase inexpiratory resistance occurs in such subjects just before the initialoccluded inspiratory effort of occlusive apnea obstructive sleep apneas.

Resistance is defined as ΔP/ΔF, where ΔP is the pressure gradient, andΔF is the flow. There is generally an increase of resistance duringtransition from wakefulness to sleep. The linear resistance at flow=0.2l/s as an accepted reference standard. However there is not available inthe art adequate literature that can relate resistance to the linearportion of the pressure flow relationship. Some researchers in the arthave reported that upper airway resistance could be infinite in patientsthat have severe narrowing or closure. This condition appears mostlywith patients having obstructive sleep apnea/hypopnea.

None of the researchers relate the flow to pressure in laminar orturbulent flow to determine the linear resistance. According to knownprinciples of fluid mechanics, linear resistance usually occurs in thelaminar region. Usually linear resistance indicates a progressivedecrease in the cross-sectional area at the linear portion of pressureflow loop. Resistance and cross-sectional area are related by the knownPoiselle equation in the linear region. Therefore it is important tofind an objective method to determine resistance.

The literature that comprises the known prior art indicates that thereis not available an objective methodology for determining resistance orinspiratory flow limitation.

It is, therefore, an object of this invention to provide a reduction inthe time required to diagnose whether a patient suffers from IFL.

It is another object of this invention to provide a methodology thatproduces an objective diagnosis of sleep apnea.

It is also an object of this invention to provide a method of diagnosingsleep apnea without requiring the use of a pressure-monitoring catheter.

It is a further object of this invention to provide a methodology thatproduces an objective determination of airway resistance in a patient.

It is additionally an object of this invention to provide a diagnosticmethodology that distinguishes between laminar and turbulent flow in theupper airway of a patient.

It is yet a further object of this invention to provide objectivecharacterization of a sleep apnea condition that will be useful incontrolling a therapy therefor.

SUMMARY OF THE INVENTION

The foregoing and other objects are achieved by this invention whichprovides a method of measuring upper airway resistance of a humanpatient. In accordance with a first method aspect of the invention,there are provided the steps of:

obtaining air pressure data from an air pressure data signalcorresponding to a plurality of breathing cycles while the human patientis asleep;

obtaining air flow data from an air flow data signal corresponding tothe plurality of breathing cycles while the human patient is asleep;

transferring the air pressure data and the air flow data to a processor;

storing the air pressure data and the air flow data in respectivecorrelated storage regions of a matrix program system of the processor;

segregating the air pressure data and the air flow data in the matrixprogram of the processor into corresponding breathing cycles of thehuman patient;

computing normalized air pressure data to achieve a predeterminednormalized air pressure value to correspond with a predetermined pointfor each breathing cycle of the human patient;

producing a correlation of the air flow data against normalized airpressure data;

curve-fitting onto the correlation of the air flow data againstnormalized air pressure data a curve corresponding to a predeterminedmultiple term mathematical function;

computing the value of the coefficients of the predetermined multipleterm mathematical function; and

computing the derivative of the predetermined multiple term mathematicalfunction.

In a specific illustrative embodiment of the invention of this firstmethod aspect there is provided in the step of curve-fitting onto thecorrelation of the air flow data against normalized air pressure data acurve, the predetermined multiple term mathematical function is aquadratic function, F(P)=AP²+BP+C, where A, B, and C are coefficients.

In a further embodiment, the step of curve-fitting onto the correlationof the air flow data against normalized air pressure data a curve, thepredetermined multiple term mathematical function is a three termpolynomial function F(P)=AP³+BP²+CP+D, where A, B, C, and D arecoefficients. The derivative of the three term polynomial functioncorresponds to the relationship:$\frac{\mathbb{d}F}{\mathbb{d}P} = {{3A\quad P^{2}} = {{2B\quad P} + {C.}}}$

There is additionally provided the step of determining that a breath isinspiratory in response to the derivative of the three term polynomialfunction having a value of zero or positive, whereby${\frac{\mathbb{d}F}{\mathbb{d}P} \geq 0}->{I\quad F\quad{L.}}$

A breath is inspiratory in response to the derivative of the three termpolynomial function having a negative value, whereby${\frac{\mathbb{d}F}{\mathbb{d}P} < 0}->{N\quad I\quad F\quad{L.}}$

The resistance is computed in response to the reciprocal of coefficientC, wherebyResistance=1/C.

In accordance with a second method aspect of the invention, there isprovided a method of determining a flow-limiting characteristic of theupper airway of a human patient, the method including the steps of:

obtaining air pressure data from an air pressure data signalcorresponding to a plurality of breathing cycles while the human patientis asleep;

obtaining air flow data from an air flow data signal corresponding tothe plurality of breathing cycles while the human patient is asleep;

transferring the air pressure data and the air flow data to a processor;

storing the air pressure data and the air flow data in respectivecorrelated storage regions of a matrix program system of the processor;

segregating the air pressure data and the air flow data in the matrixprogram of the processor into corresponding breathing cycles of thehuman patient;

computing normalized air pressure data to achieve a predeterminednormalized air pressure value to correspond with a predetermined pointfor each breathing cycle of the human patient; and

computing the flow-limiting characteristic of the upper airway of ahuman patient as a function of normalized air pressure data divided bycorresponding air flow data.

In a specific illustrative embodiment of the invention of this secondmethod aspect, the matrix program system is a spreadsheet programsystem, and the air pressure data and the air flow data being arrangedin respective spreadsheet columns correlated by rows. Normalized airpressure data is stored in a respective spreadsheet column correlated byrows into corresponding breathing cycles of the human patient.

Each breathing cycle of the human patient is determined in relation tothe predetermined point thereof corresponding to the predeterminednormalized air pressure value. Generally, inspiration precedesexpiration, and flow is zero at the commencement of the breath cycle. Inaddition, the predetermined normalized air pressure value corresponds toa zero value. There is further provided the step of computing theflow-limiting characteristic of the upper airway of a human patient foreach of the plurality of breathing cycles. The air pressure data and theair flow data are sampled a plurality of times during each breathingcycle. Also, the step of computing the flow-limiting characteristic ofthe upper airway of a human patient is performed a correspondingplurality of times during each breathing cycle. The step of computingthe flow-limiting characteristic of the upper airway of a human patientis performed a corresponding plurality of times during each breathingcycle and during which the air flow data has a predetermined value. Theair flow data and the normalized pressure data are correlated to form adata correlation in a data correlation array, and the predeterminedvalue of the air flow data is determined within a substantially linearportion of the data correlation. The predetermined value of the air flowdata is approximately between 0.00 L/s and 0.22 L/s, and thepredetermined value of the air flow data is approximately 0.20 L/s.

In a further embodiment, there is provided the further step of computinga slope of the correlated air flow data and normalized pressure datawithin the substantially linear portion of the data correlation. A dataarray is produced corresponding to the flow-limiting characteristicwherein the normalized air pressure data corresponds to the x-axis andthe air flow data corresponds to the y-axis.

In accordance with a third method aspect of the invention, there isprovided a method of measuring upper airway resistance of a humanpatient, the method is provided with the steps of:

obtaining air pressure data from an air pressure data signalcorresponding to a plurality of breathing cycles while the human patientis asleep;

obtaining air flow data from an air flow data signal corresponding tothe plurality of breathing cycles while the human patient is asleep;

transferring the air pressure data and the air flow data to a processor;

storing the air pressure data and the air flow data in respectivecorrelated storage regions of a matrix program system of the processor;

segregating the air pressure data and the air flow data in the matrixprogram of the processor into corresponding breathing cycles of thehuman patient;

computing normalized air pressure data to achieve a predeterminednormalized air pressure value to correspond with a predetermined pointfor each breathing cycle of the human patient;

producing a correlation of the air flow data against normalized airpressure data;

curve-fitting onto the correlation of the air flow data againstnormalized air pressure data a curve corresponding to a three termpolynomial function F(P)=AP³+BP²+CP+D, where A, B, C, and D arecoefficients; and

computing the value of the upper airway resistance of a human patient asan inverse function of coefficient C of the three term mathematicalfunction, C, wherebyResistance=1/C.

Each breathing cycle is defined at an onset point where inspiratory flowis zero. Moreover, each breathing cycle is defined at an onset pointwhere supraglottic pressure has been normalized to zero. The derivativeof the three term polynomial function is computed in accordance with therelationship:$\frac{\mathbb{d}F}{\mathbb{d}P} = {{3A\quad P^{2}} + {2B\quad P} + {C.}}$

Then, the presence of inspiratory flow limitation is determined inresponse to the derivative of the three term polynomial function.

In a specific illustrative embodiment of the invention, the step ofobtaining air flow data from an air flow data signal includes thefurther step of recording the air flow data signal on a polygraph.Additionally, the step of storing the air pressure data and the air flowdata in respective correlated storage regions of a matrix program systemof the processor include the step of exporting the air pressure data andthe air flow data to a first Excel® spreadsheet. Then, a graphicalrepresentation of adjusted time along the x-axis and flow along they-axis is plotted. There is then performed a step of curve fitting aninspiratory rising limb flow-time curve to a mathematical polynomialfunction F(P)=A t³+B t²+Ct+D. where A, B, C, and D are coefficients. Thederivative of the mathematical polynomial function is then calculated,and the value of the derivative of the mathematical polynomial functionis exported to a second Excel® spreadsheet. Whether a breath is or isnot flow limited is determinwed in response to the value of thederivative of the mathematical polynomial function.

BRIEF DESCRIPTION OF THE DRAWING

Comprehension of the invention is facilitated by reading the followingdetailed description, in conjunction with the annexed drawing, in which

FIG. 1 is a graphical representation that shows pressure-flow loopsillustrating NIFL and IFL breaths;

FIGS. 2A and 2B are graphical representations that illustrate themathematical nature of a polynomial function (FIG. 2A) and a quadraticfunction (FIG. 2B).

FIGS. 3A, 3B, 3C, and 3D are graphical representations that illustratethe hypothesized considerations regarding the ability of the polynomialand quadratic functions to distinguish between NIFL (FIGS. 3A and 3B)and IFL breaths (FIGS. 3C and 3D);

FIGS. 4A, 4B, 4C, and 4D are graphical representations that illustratethe sequences of the analyses;

FIGS. 5A and 5B are graphical representations that illustrate theanalyses conducted as described under Protocol 1 herein; and

FIGS. 6A and 6B are graphical representations that show IFL breath andthe fitted hyperbolic function when flow data is fitted to raw pressuredata and when the data is fitted to pressure data that has beentransformed to the absolute value.

DETAILED DESCRIPTION

In developing the foundations of the present invention, the inventorsherein consider a steady homogenous flow in a circular cylinder (theupper airway), with the assumption that the flow of air in the upperairway will expand without the loss or gain of heat. Consider astreamline of air, which connects two points M₁, the upstream pressure,which is atmospheric pressure in the present model, and M₂, thedownstream pressure, which is equivalent to supraglottic pressure in thepresent model. For each point, there is a density (ρ), pressure (P),area (A), velocity (V) and flow (F) that characterize that point. In themodeling that follows, it should be noted that the goal is determinationof the flow of the upper airway at the downstream pressure point, M₂.Flow, which is constant throughout the upper airway, is given by:Total Energy=Kinetic Energy [T]+Potential Energy [E]+InternalEnergy  (1)

Internal energy=0

Differentiate both sides of equation (1) $\begin{matrix}{{dK} = {{dT} + {dE}}} & (2) \\{{T = {\frac{1}{2}m\quad V^{2}}},{E = {pv}}} & (3)\end{matrix}$

The gas unit mass (m) is defined according equation (4) $\begin{matrix}{{m = {{\rho\quad v} = 1}},{v = \frac{1}{\rho}},{{d\quad v} = {d\left( \frac{1}{\rho} \right)}}} & (4) \\{{d\quad K} = {{V\quad d\quad V} + {P\quad d\quad v}}} & (5) \\{{{\frac{1}{2}V^{2}} + {\int_{\rho_{0}}^{\rho_{00}}{P\quad{\mathbb{d}\left( \frac{1}{\rho} \right)}}}} = {const}} & (6)\end{matrix}$Integrating by part the second term $\begin{matrix}{{{\frac{1}{2}V^{2}} + \frac{P}{\rho_{0}} - \frac{P}{\rho_{0}} - {\int_{\rho_{0}}^{\rho}\frac{\mathbb{d}P}{\rho}}} = {const}} & (7)\end{matrix}$  Since the path is short then ρ₂≅ρ₁=ρ  (8)Rearrange and substitute P=Kρ^(γ)(a), dP=Kγρ^(γ−1)  (8b)Substitute (8b) in (7)One obtains $\begin{matrix}{{{\frac{1}{2}V^{2}} + \frac{P}{\rho} - \frac{P}{\rho_{0}} - {\left( \frac{1}{\gamma - 1} \right)\left\lbrack {\frac{K\quad\rho_{0}^{\gamma}}{\rho_{0}} - \frac{K\quad\rho^{\gamma}}{\rho}} \right\rbrack}} = {const}} & (9)\end{matrix}$The terms in equation (9) can be reduced as${\frac{P}{\rho_{0}} \cong {\frac{P}{\rho}\quad{and}}},{\frac{K\quad\rho_{0}^{\gamma}}{\rho_{0}} = {const}}$ Continuity equation F=ρ ₁ A ₁ V ₁=ρ₂ A ₂ V ₂  (10)Solving for V₁: $\begin{matrix}{V_{1} = {{\frac{\rho_{2}A_{2}}{\rho_{1}A_{1}}V_{2}} = {\Omega\quad V_{2}}}} & (11)\end{matrix}$Where, $\begin{matrix}{\Omega = {\frac{\rho_{2}A_{2}}{\rho_{1}A_{1}} = \frac{A_{2}}{A_{1}}}} & (12)\end{matrix}$

The Bernoulli or energy equation for homogenous fluid such as air, onone streamline, through M₁, M₂ and neglecting the effect of gravity is:$\begin{matrix}{{\frac{P_{1}}{\rho_{1}} + {\frac{1}{2}V_{1}^{2}}} = {\frac{P_{2}}{\rho_{2}} + {\frac{1}{2}V_{2}^{2}}}} & (13)\end{matrix}$Because air is a compressible, consideration needs to be given to theheat kinematics ratio $\frac{\gamma}{\gamma - 1}.$If the kinematics heat ratio is set as:${K = \frac{\gamma}{\gamma - 1}},$then equation (13) can be rewritten as derived by equation (9), as:$\begin{matrix}{{{K\frac{P_{1}}{\rho_{1}}} + {\frac{1}{2}V_{1}^{2}}} = {{K\frac{P_{2}}{\rho_{2}}} + {\frac{1}{2}V_{2}^{2}}}} & (14)\end{matrix}$Because the path of the upper airway is short it may be assumed thatρ₁≅ρ₂=ρ. Then equation 4 is rearranged as: $\begin{matrix}{{P_{1} - P_{2}} = {\frac{\rho}{2K}\left( {V_{2}^{2} - V_{1}^{2}} \right)}} & (15)\end{matrix}$Substituting V₁ ² from equation 2: $\begin{matrix}{{P_{1} - P_{2}} = {\frac{\rho}{2K}\left( {V_{2}^{2} - {\Omega^{2}V_{2}^{2}}} \right)}} & (16)\end{matrix}$Solving for V₂ ²: $\begin{matrix}{V_{2}^{2} = {2K\frac{\left( {P_{1} - P_{2}} \right)}{\rho\left( {1 - \Omega^{2}} \right)}}} & (17)\end{matrix}$Squaring both sides of equation 1, to obtain the flow squared at pointM₂:F ²=ρ₂ A ₂ ² V ₂ ²  (18)Substituting for V₂ ² from equation 7: $\begin{matrix}{F^{2} = {\frac{2\rho\quad A_{2}^{2}K}{\left( {1 - \Omega^{2}} \right)}\left( {P_{1} - P_{2}} \right)}} & (19)\end{matrix}$Rearranging: $\begin{matrix}{F^{2} = {\frac{2\rho\quad A_{2}^{2}K}{\left( {1 - \Omega^{2}} \right)}{P_{1}\left( {1 - \frac{P_{2}}{P_{1}}} \right)}}} & (20)\end{matrix}$Taking the square root of both sides of equation 10, then one obtains$\begin{matrix}{{F = {\left( \frac{2\rho\quad A_{2}^{2}{KP}_{1}}{\left( {1 - \Omega^{2}} \right)} \right)^{\frac{1}{2}}\left( {1 - \frac{P_{2}}{P_{1}}} \right)^{\frac{1}{2}}}}{Let}{G = \left( \frac{2\rho\quad A_{2}^{2}{KP}_{1}}{\left( {1 - \Omega^{2}} \right)} \right)^{\frac{1}{2}}}} & (21)\end{matrix}$Therefore, flow through a streamline between two points, M₁ and M₂, isgiven by: $\begin{matrix}{F = {G\left( {1 - \frac{P_{2}}{P_{1}}} \right)}^{\frac{1}{2}}} & (22)\end{matrix}$Using Newton's expansion law:$\left( {1 + X} \right)^{N} = {1 + {NX} + {\frac{N\left( {N - 1} \right)}{2!}X^{2}} + {\frac{{N\left( {N - 1} \right)}\left( {N - 2} \right)}{3!}X^{3}} + \ldots}$One obtains: $\begin{matrix}{F = {G + {\frac{G}{2P_{1}}P_{2}} + {\frac{G}{8P_{1}^{2}}P_{2}^{2}} + {\frac{3G}{48P_{1}^{3}}P_{2}^{3}} + \ldots}} & (23)\end{matrix}$Letting,${A = \frac{3G}{48P_{1}^{3}}},{B = \frac{G}{8P_{1}^{2}}},{C = \frac{G}{2P_{1}}},{D = G}$One then can substitute these coefficients into equation 23 to get apolynomial function that approximates flow (F) in terms of thesupraglottic pressure. For this function, it is assumed that P₁ isatmospheric pressure, which is a constant, and P₂=P, which now isdefined as the supraglottic pressure:F=AP ³ +BP ² +CP+D  (24).

Per Newton's expansion law, the relationship between pressure and flowcould also be predicted by a quadratic equation:F=AP ² +BP+C  (25).

From the nature of a polynomial function the inventors predicted that apolynomial function would be expected to provide a better estimate ofthe pressure-flow relationship than can be achieved with the quadraticfunction for flow-limited breaths. This results from the fact that, forIFL breaths, the polynomial function is characterized by twodeflections, as illustrated in FIG. 2. A two deflection relationshipwill more closely approximate the measured pressure-flow relationship ofIFL breaths, which are characterized by a point of maximum flow,followed by a decrease and plateau in flow, as shown in FIG. 1.

FIG. 1 is a graphical representation that shows pressure-flow loopsillustrating NIFL and IFL breaths. A breath was labeled IFL if there wasa ≧1 cmH₂O decrease in supraglottal pressure (P_(SG)) without anycorresponding increase in flow (V) during inspiration.

The quadratic function, on the other hand, is characterized by only onedeflection, as shown in FIG. 2B. FIGS. 2A and 2B are graphicalrepresentations that illustrate the mathematical nature of a polynomialfunction (FIG. 2A) and a quadratic function (FIG. 2B). As shown, thepolynomial function is characterized by two deflections, Min and Max,whereas the quadratic function is characterized by one deflection, Max.

While performing the initial curve-fitting analysis, as will bedescribed in greater detail below, it is noted that the nature of thepolynomial function, in contrast to the quadratic function, allows forthe objective differentiation of IFL and NIFL breaths. In particular, itis noted that for the polynomial function, the maximal flow of thepredicted relationship usually is located at the correspondingly samepoint as the measured maximal flow. In contrast, the predicted maximalflow for the quadratic function is at a more negative pressure. Theinventors herein have hypothesized that a more objective result can beachieved by determining the presence of flow-limitation by examiningderivative of the polynomial function. This would correspond to theslope of the pressure-flow relationship. The derivative of thepolynomial function is: $\begin{matrix}{\frac{\mathbb{d}F}{\mathbb{d}P} = {{3{AP}^{2}} + {2{BP}} + C}} & (26)\end{matrix}$

FIGS. 3A, 3B, 3C, and 3D are graphical representations that illustratethe hypothesized considerations regarding the ability of the polynomialand quadratic functions to distinguish between NIFL (FIGS. 3A and 3B)and IFL breaths (FIGS. 3C and 3D). The vertical straight line in all ofthese figures is located at the measured maximum flow (FIGS. 3A and 3C).There are shown in these figures the measured pressure-flow relationship(solid line) and the hypothesized polynomial (dashed line), and thequadratic (dash-dot line) relationships. In FIGS. 3B and 3D, there areshown the slopes of the predicted functions at increasing values ofP_(SG). The slope at the measured maximal flow for both, the polynomialand the quadratic function, remains negative for NIFL breaths (FIG. 3B).The slope of the polynomial function at the measured maximal flowbecomes positive for IFL breaths, whereas the slope of the quadraticfunction remains negative (FIG. 3D).

Theoretically, for non-flow limited breaths, flow would continue toincrease beyond the point of maximal flow ifthere were further decreasesin supraglottic pressure. Therefore, the derivative of the polynomialfunction (or the slope of the pressure-flow curve) at the actual maximalflow is negative. This is illustrated in FIG. 3A which shows a NIFLbreath and the theoretic relationship using the polynomial function. Atthe measured maximal flow, the slope of the theoretic pressure-flowrelationship is negative, as illustrated in FIG. 3B. However, forbreaths that demonstrate inspiratory flow limitation, there are nofurther increases in flow despite decreasing supraglottic pressure (FIG.3C). Therefore, the slope or derivative of the polynomial function atthe measured maximal flow is either zero or positive for flow-limitedbreaths (FIG. 3D). Therefore, at maximal flow, two cases can bedetermined from equation 26, as follows:if ${{\left. 1 \right)\quad\frac{\mathbb{d}F}{\mathbb{d}P}} < 0},$the breath is non-flow-limited; andif${{{\left. 2 \right)\quad\frac{\mathbb{d}F}{\mathbb{d}P}} > {0\quad{or}\quad\frac{\mathbb{d}F}{\mathbb{d}P}}} = 0},$the breath is flow-limited.

By a similar analysis the inventors hypothesized that the derivative ofthe quadratic function cannot be used to determine if the pressure-flowrelationship demonstrates flow limitation. The derivative of thequadratic function is given as: $\begin{matrix}{\frac{\mathbb{d}F}{\mathbb{d}P} = {{2{AP}} + B}} & (27)\end{matrix}$However, if the quadratic function is used to characterize thepressure-flow relationship (FIGS. 3A and 3C), the derivative of thequadratic function cannot be used to distinguish between non-flowlimited and flow-limited breaths. This is illustrated in FIGS. 3B and3D, which shows that the derivative of the quadratic equation will benegative for both types of breaths. In other words,$\frac{\mathbb{d}F}{\mathbb{d}P} < {0\quad{for}\quad{all}\quad{{breaths}.}}$

In summary, theoretical considerations indicate that the relationshipbetween flow and supraglottic pressure in the upper airway can becharacterized by either a quadratic or polynomial function. However,based upon the theoretical considerations, the polynomial function wasthe better of the two functions to model the upper airway mathematicallybecause it would provide the best fit compared to the actualpressure-flow relationship. Its derivative provides an objective andaccurate methodology for the detection of inspiratory flow limitation.

In determining linear resistance, it is first understood that flowgenerally will proceed from an area of high pressure to an area oflow-pressure, and that when the velocity of the air increases, the flowwill change from laminar to turbulent. This is consistent with Bernoulliprinciple. In the laminar flow condition, the relation between pressureand flow is linear, and in the turbulent flow condition, therelationship is relatively non-linear. Therefore, in order to quantifythe linear resistance, one needs to determine the flow in the laminarregion, in accordance with the following principle for viscous flow:

F∝P^(N), where F is the flow, P is the pressure, and N is an exponent.If N>1, or N<1 then the flow is turbulent. If N=1 then the flow islaminar.

Referring once again to polynomial equation F(P)=AP³+BP²+CP+D, it can beseen that this equation has two phases of flow, i.e., laminar andturbulent. The equation is then recharacterized as follows:F(P)=AP ³ +BP ² +CP+D=F ₁ +F ₂Where F₁=AP³+BP² for turbulence flow, and F₂=CP+D for laminar Flow. Thereciprocal slope of the laminar flow is the first linear resistance:$R = {\frac{1}{C} = {\frac{\Delta\quad P}{\Delta\quad F}.}}$Thus, the polynomial coefficient C is used to determine the linearresistance.Methodologies

The methodologies and corresponding data analyses in accordance with thepresent invention are organized into several major steps. The first stepis the curve fitting. In order to determine the best function foreffecting a correlation with measured results, actual data was curve fitby the inventors using five different mathematical functions. Aselection then is made of the function that has the highest correlationR². In the second step, an error fit method is selected to achieve theleast error fit. Finally, the analytical hypothesis is used tocharacterize the air flow. All these steps are performed on 50 breaths.At the final stage of the process, sensitivity analyses are performed todetermine which function can best model the experimental data orpressure-flow loop.

Measurements and Manual Determination of Flow Limitation

For each breath, airflow (V) was measured by a pneumotachometer (Model3700A, Hans Rudolph, Inc.) attached to a nasal mask. Supraglottic airwaypressures were measured using a pressure-tipped catheter (ModelTC-500XG, Millar Co.) threaded though the mask and positioned in theoropharynx just below the base of the tongue. Correct placement wasverified by visually inspecting the catheter's position in theoropharynx.

FIGS. 4A, 4B, 4C, and 4D are graphical representations that illustratethe sequences of the analyses. As shown in FIG. 4A, three breaths arerepresented from the raw tracing of a polygraph (not shown). Theanalysis herein was performed on the middle (boxed) breath. FIG. 4Bshows the pressure loop of the indicated breath. The selected breathindicates flow limitation because there is no increase in flow despitea >1 cmH₂O increase in P_(SG). FIG. 4C illustrates the curve fittinganalysis, showing only the ascending limb of the inspiratory portion ofthe pressure-flow loop (solid line). The equation for the fitted curveis:F(P)=−0.0005P³−0.0151P ²−0.0302P−0.1137.Because the slope of the polynomial function is 0.001 (i.e., >0) atmeasured maximal flow (vertical line), the breath is characterized asIFL by the model. The EEG is the electroencephalogram, and SaO₂ is thearterial oxygen saturation.

During the studies, airflow and supraglottic pressure were recordedsimultaneously with Biobench data acquisition software (NationalInstruments, Austin, Tex.) on a separate computer (FIG. 4A). For eachbreath, the onset of inspiration was defined as the sampling point atwhich V₁=0. In response to the rare occurrence where there was a shiftin baseline, the nadir flow was determined and the flow values shiftedappropriately. Because the Miller catheter provides relative pressures,PSG was set to zero for the inspiration onset sampling point and theremaining values for the breath were calculated. A pressure flow loopwas generated (FIG. 4B) and the loop was analyzed for the presence ofinspiratory flow limitation (IFL) (FIG. 1). A breath was labeled to beIFL if there was a ≧1 cmH₂O or greater decrease in supraglottic pressurewithout any corresponding increase in flow during inspiration. If theflow-pressure relationship did not meet this criterion, the breath waslabeled as non-flow limited (NIFL).

All analyzed breaths in the following protocols were obtained duringStage 2 NREM sleep. Breaths from wakefulness were not analyzed, as IFLis not observed during wakefulness. As slow wave and REM sleep are nottypically observed in the heavily instrumented subjects, breaths duringthese stages of sleep were not available for analysis. In addition, onlybreaths free from artifact were included in the analysis. All breathswere obtained from healthy polysomnography. The demographics of thesubjects were presented within each protocol. Non-complaining adults whohad volunteered for research studies in the laboratory were used assubjects. All subjects were free of sleep-disordered breathing, asmeasured by apneas and hypopneas, on baseline.

The first inquiry is whether the polynomial function best predicts therelationship between pressure and flow in the upper airway.

Step 1; Curve Fitting—First, the inventors performed a curve fittingstep to model the upper airway mathematically. Sigma Stat 2.0 software{(FIGS. 4C and 5A)} was used in the analysis, the point of which was todetermine which of following five regression equations (Table 1) bestestimated inspiratory flow (the dependent variable) as a function ofsupraglottic pressure (the independent variable). This process issimilar to performing a linear regression, in which the predictedrelationship can be given by the equation: F(P)=AP+B. TABLE 1 FUNCTIONSUSED FOR CURVE FITTING One-term hyperbolic F(P) = AP/(B + P) Two-termhyperbolic F(P) = AP/(B + P) + CP/(D + P) + FP Exponential F(P) =Ae^(−BP) + Ce^(−DP) Quadratic F(P) = AP² + BP + C Polynomial F(P) =AP³ + BP² + CP + DF(P), flow as a function of pressure;A, B, C, D, and F, coefficientse, exponential mathematical constant (˜2.78)However, since the pressure-flow relationship is not linear, theinventors herein used five non-linear regression functions. The firsttwo are derived from the theoretical considerations above: quadratic andpolynomial. The third, a single-term hyperbolic, has previously beenproposed as an accurate predictor of the pressure-flow relationship. Inaddition, the inventors herein analyzed two additional functions:double-term hyperbolic and exponential. Neither the pressure nor flowvalues were transformed prior to the curve fitting. This analysis wasperformed on 20 breaths, 10 NIFL, 10 IFL derived from 4 subjects (1male, 3 females, mean age 22±3 yrs, mean BMI 23.0±3.0 kg/m²). For eachcalculated function, the inventors herein determined the coefficient ofdetermination (R²), which indicates how much of the variability in onevariable (flow) is explained by knowing the value of the other(supraglottic pressure) (12). The R² for IFL and NIFL breaths werecompared between the five functions using one-way repeated measuresanalysis of variance (ANOVA), with breath number as the repeated measureand the function as the factor for comparison. If there was asignificant difference between the groups, a Student-Newman-Keuls testwas performed to detect between group differences with P<0.05 set as thelevel for a significant test. The same test was performed on thecombined groups of breaths.

Step 2: ErrorFit: To determine the degree of approximation between thepressure-flow relationship derived from either the quadratic orpolynomial function to the actual pressure-flow relationship, theinventors herein determined the error-fit for 50 breaths, 25 each NIFLand IFL derived from 8 subjects (5 males, 3 females, mean age 25±4years, mean BMI 26.2±4.8 kg/m²). Only the quadratic and polynomialfunctions were studied based upon the results of the curve fittinganalysis. An illustration of the concept of error-fit is given in FIGS.5A and 5B.

FIGS. 5A and 5B are graphical representations that illustrate theanalyses conducted as described under Protocol 1 herein. FIG. 5A is anexample of curve fitting that shows the actual data points (∘) and thepredicted pressure-flow relationships if the points are fitted to aquadratic function (solid line) or the two-term hyperbolic function(dashed line). FIG. 5B illustrates an example of error fit that showsthe actual (solid line) and predicted (dashed line) pressure-flowrelationships. The predicted relationship uses the quadratic function.The shaded area is the graphical representation of the mathematicalformula for error fit.

As noted, FIG. 5B shows the actual pressure-flow relationship for an IFLbreath (solid line) and the predicted pressure-flow relationship usingeither the quadratic function (dashed line). The gray-shaded areas showthe difference between the two relationships. The smaller thegray-shaded area, the smaller the error-fit and the more closely thepredicted relationship approximates the actual pressure-flowrelationship. The error-fit is a mathematical representation of thisgray-shaded area. Mathematically, error fit is defined as:$\begin{matrix}{100\left( {{\overset{k}{\sum\limits_{i}}1} - \left( {y_{k} - y_{i}} \right)} \right.} & (18)\end{matrix}$where $\sum\limits_{i}^{k}$is the summation of a series of points, y_(k) represents the points inthe original function and y_(i) represents the points in the fittedfunction. Using this formula, as the predicted pressure-flowrelationship more closely approximates the actual relationship, theerror-fit or difference between the two relationships decreases. Theerror-fit for IFL and NIFL breaths were compared between the fivefunctions using one-way repeated measures analysis of variance (ANOVA),with the breath number as the repeated measure and the function as thefactor for comparison. If there was present a significant differencebetween the groups, a Student-Newman-Keuls test was performed to detectthe differences between the groups with P<0.05 set as the level for asignificant test. The same test was performed on the combined groups ofbreaths.

The next question to be considered is whether the polynomial functionobjectively detects flow limitation?

Using the same 50 breaths on which the inventors herein determined theerror-fit, the inventors herein determined the slope at the measuredmaximal flow for the polynomial equation. Per the hypothesis, if theslope at the measured maximal flow was <0, the inventors herein labeledthe breath NIFL; if the slope at the measured maximal flow was ≧0, theinventors herein labeled the breath IFL. The inventors calculated thesensitivity, specificity, positive predictive value (PPV) and negativepredictive value (NPV) for the detection of IFL breaths by thepolynomial model compared to the standard method (described at thebeginning of the Methods section) using standard formulas.

To confirm the hypothesis that the slope at the measured maximal flowfor the quadratic equation would be negative for both IFL and NIFLbreaths, the inventors herein determined the slope at the measuredmaximal flow for the same 50 breaths. The inventors herein report theproportion of NIFL and IFL breaths with a negative slope.

To validate the results, the inventors herein then determined the slopeat the measured maximal flow using the polynomial equation for 544randomly selected breaths from 16 subjects without sleep-disorderedbreathing as measured by apneas and hypopneas (10 males, 10 females,mean age 30±8 yrs, mean BMI 25.2±4.3 kg/m²). Applying the hypothesis,the inventors herein labeled each breath as NIFL or IFL. Thesensitivity, specificity, positive predictive value (PPV) and negativepredictive value for the detection of IFL breaths are calculated by thepolynomial model compared to the standard method using standardformulas.

Results

Protocol 1

The results showed that the polynomial and quadratic functions hadbetter fits to the data than the single- and double-term hyperbolic andexponential functions. However, when using a test that determines thedegree of correlation between the actual and experimental relationships(error-fit), only the polynomial function accurately predicts thepressure-flow relationship.

Sensitivity and specificity analyses in the development stage werehigher for polynomial function than quadratic function using thederivative of each function. Therefore the inventors determined thatpolynomial function should be used for final validation of themathematical models.

Curve Fitting: The results of the curve fitting are presented in Table2. There was a significant difference between the R² values when all thebreaths are combined and for the NIFL and IFL breaths when analyzedseparately (P<0.001 for all three comparisons). TABLE 2 R² VALUES FORTHE VARIOUS FUNCTIONS Quadratic Polynomial Single-HyperbolicDouble-Hyperbolic Exponential IFL Breaths 0.85 ± 0.10  0.90 ± 0.060 0.61± 0.14 0.79 ± 0.13 0.55 ± 0.32 NIFL Breaths 0.89 ± 0.06 0.92 ± 0.04 0.54± 0.20 0.70 ± 0.24 0.79 ± 0.24 All Breaths 0.88 ± 0.08 0.91 ± 0.05 0.57± 0.17 0.78 ± 0.19 0.67 ± 0.30Values are means ± SE;IFL, inspiratory flow limited;NIFL, non-inspiratory flow limited

For NIFL breaths, post-hoc testing showed that R² was significantlylarger for the polynomial function compared to all other functions andthat the quadratic function had a larger mean R² compared to other threefunctions. For IFL breaths, there was no difference in the mean R²values between the quadratic, polynomial and double hyperbolicfunctions. All three functions had larger mean R² values compared to thesingle-hyperbolic and exponential functions. For all the breathscombined, the mean R² was highest for the polynomial function. Inaddition, the R² values were higher for the quadratic equation comparedto the other three functions. In summary, the polynomial and quadraticfunctions had better fits to the data than the single- and double-termhyperbolic and exponential functions. Therefore, further analysis wasperformed only on the quadratic and polynomial functions.

Error-Fit: Representative graphs depicting the relationship between theactual pressure-flow curve and the curve as predicted by either thequadratic or polynomial equations for one IFL and one NIFL breath isillustrated in FIG. 3. As can be seen, there is more overlap (lesserror) between the actual and predicted curves for the polynomialfunction than for the quadratic function. For the total group of 50breaths, the error fits for the polynomial function were smaller onaverage than the quadratic function for the IFL breaths (2.0≦2.7% vs.25.0±22.2%, P<0.001), NIFL breaths (4.0±7.7% vs. 16.0±14.0%, P=0.003)and for all breaths combined (3.3±0.06% vs. 21.1±19.0%, P<0.001).

In summary, the curve-fitting of the pressure-flow relationship in theupper airway will result in a tight fit (high R²) of the data only forthe quadratic and polynomial functions. However, when using a test thatdetermines the degree of correlation between the actual and experimentalrelationships (error-fit), only the polynomial function accuratelypredicts the pressure-flow relationship.

Protocol 2

Step 1: The sensitivity, specificity, PPV and NPV for the detection offlow limitation in the initial 50 breaths using the polynomial functionis summarized in Table 3. TABLE 3 SENSITIVITY/SPECIFICITY ANALYSISDevelopment Breaths Validation Breaths (n = 50) (n = 544) Sensitivity100 99 Specificity 100 99 PPV 100 97 NPV 100 99PPV, positive predictive value;NPV, negative predictive value

As the table illustrates, the use of the slope at maximal flow of thepolynomial equation results in both high sensitivity and specificity forthe determination of IFL breaths. PPV and NPV were also high. For thequadratic function, the inventors herein have confirmed that themajority of breathes of both NIFL (24 of 25, 96%) and IFL (22 of 25,88%) IFL breaths had a negative slope, indicating that the quadraticfunction would be unhelpful in detecting IFL breaths.

Step 2: In the larger group of breaths, sensitivity and specificityremained high (Table 3, right column), as did the PPV and NPV.

In summary, in Protocol #2, a sensitivity/specificity analysis of theuse of polynomial function was performed to detect IFL breaths comparedto the standard method using a pressure-flow loop. This analysisindicates that the polynomial function has an excellent ability topredict the presence of flow-limitation in the pressure-flowrelationship. In contrast, the quadratic function cannot be used todistinguish between IFL and NIFL breaths.

Findings

There are three major findings of this analysis. First, a polynomialequation, F(P)=AP³+BP²+CP+D, provides an estimation of the upper airwaypressure-flow relationship with relative precision compared to othermathematical equations. Second, the derivative of this equation can beused to objectively and precisely determine the presence of inspiratoryflow limitation. The Coefficients A, B, C, and D are part of thepolynomial function and serve to identify the mathematical relation shipbetween pressure and flow or between flow or time. The A and Bcoefficients serve to describe principally the breath in turbulence, andthe coefficients C and D describe laminar flow at low velocity. Thederivative of function serves to specify the type of breath,particularly as to whether or not it is flow limited. Third, thecoefficient C is useful to determine the linear resistance. The mainrequirement for the accurate determination of IFL using the polynomialfunction is a continuous and simultaneous measurement of flow andsupraglottic pressure.

The relationship between flow and pressure in the upper airway duringwakefulness was first described by Rohrer using the equation:P=K₁*V+K₂*V², where *V is flow and K_(1 and K) ₂ are constants. Ahyperbolic function (see Table 1) was shown to characterize better theupper airway pressure-flow relationship during sleep, as indicated by acorrelation coefficient of 0.89 compared to 0.55 for the Rohrerequation. The characterization was better because the hyperbolicequation approximated the pressure-flow relationship for both NIFL andIFL breaths. Similarly, others recently found that the hyperbolicequation better characterized by the pressure-flow relationship, asevidenced by larger Pearson's square correlations for all breathsanalyzed as well as for the subset of IFL breaths. In contrast, theinventors herein have found that a 3-term polynomial function bestcharacterized the pressure-flow relationship during sleep. In addition,a hyperbolic function provided a poor characterization of thepressure-flow relationship.

The importance of the use of the three term function is illustrated inFIGS. 6A and 6B. FIGS. 6A and 6B are graphical representations that showIFL breath and the fitted hyperbolic function when flow data is fittedto raw pressure data and when the data is fitted to pressure data thathas been transformed to the absolute value. It is to be noted that inFIG. 6A, a hyperbolic curve provides a relatively poor representationfor the actual flow relationship, whereas in FIG. 6B, the hyperboliccurve provides a reasonable representation of the pressure-flowrelationship.

As can be seen from these figures, if positive values are used forpressure values, a hyperbolic curve closely approximates the actualpressure-flow relationship (FIG. 6A). The inventors nevertheless assertthat the use of negative values for pressure is proper because themathematical equations for curve fitting were derived to determine therelationship between the predicted and observed (actual), nottransformed variables.

Limitation

The hypothesis hereinabove presented has a potential limitation,particularly in the application of Newton's expansion law. The inventorsformulated a constant G that contains multiple parameters includingdensity, area, atmospheric pressure, and kinematics heat ratio.Therefore, for G to be constant, these parameters must be assumed alsoto be constants during the flow between M₁ and M₂. The assumption thatdensity and kinematic heat ratio are constants is based uponthermodynamic principles. It is believed that G is constant during anygiven breath, and the excellent agreement between the measured data, andpolynomial function data supports the validity of this and otherassumptions hereinabove set forth.

To ascertain the accuracy of mathematical detection of IFL, theinventors needed a “benchmark” for detection of flow limitation. Anarbitrary degree of dissociation between pressure and flow for a 1 cmdecrement in supraglottic pressure. However, the physiologicconsequences of such mild degree of inspiratory flow limitation are notknown. Conversely, mathematical methods and visual methods wereremarkably reproducible indicating that this choice of parameter isvalid for the recognition of the phenomenon. Accordingly, the presentinvestigation provides an objective operational definition that can beused in future studies to ascertain physiologic relevance.

Inspiratory flow limitation in the present study was evaluated as adichotomous variable. However, deviation from linearity between flow andpressure is a continuous variable. The present method detects flowlimitation as defined by a plateau in flow only. Any other linear flowprofile is classified as non-flow limitation. It can be argued thatchanges in the slope of the pressure-flow relationship indicatepharyngeal narrowing and turbulent flow. In fact, these were the breathsmissed by the mathematical equation. However, it is doubtful that thereis a physiologic significance of deviation from linearity without trueflow limitation.

Finally, detection of inspiratory flow limitation in the present studyrequired the use of supraglottic pressure measurement via a pharyngealcatheter and quantitative flow measurement using a sealed mask and apneumotachometer. This combination is rather intrusive and may not befeasible for routine clinical use. As noted below, IFL can be detectedfrom the flow versus time profile.

As noted, the percentage of breaths that are flow limited is related toBMI, upper airway resistance, and the presence of long-termfacilitation. Therefore, a determination of the presence offlow-limitation is expected to provide an alternative metric to assessthe relationship between SDB and daytime consequences such as excessivedaytime sleepiness and cardiovascular morbidity, particularly innon-apneic forms of the syndrome.

Non-invasive Approach

The inventors herein have established that there is a non-invasiveaspect to the present invention. More specifically, the resistance inthe Upper Airway is based on time flow. This is derived from the samehypothesis applied hereinabove in regard of the polynomial time flowfunction.

Since the flow is adiabatic, the first law of ideal gas at the site ofsupraglottic pressure (i.e., a mixed gas at constant temperature states)permits application of the thermodynamic Dalton model, as follows:$\begin{matrix}{\frac{P_{1}}{P_{2}} = {\frac{V_{1}}{V_{2}}\text{:}}} & (a)\end{matrix}$pressure is proportional to volume, V is volume and Compressible FlowPolytrobic Cycle with constant specific heat or air: $\begin{matrix}{{{PV}^{\quad M} = C}{{{{where}\quad M} = \left( {{\pm 1},{\pm 1.3}} \right)},\quad{{{for}\quad{ideal}\quad{gas}\quad M} = 1},{{{therefore}\quad P} = {CV}^{- 1}}}{V = {{\int{F{\mathbb{d}t}}} = {{{Ft}\quad ©\quad\text{)}\quad{Volume}} = {{flow}\quad{by}\quad{time}}}}}} & (b)\end{matrix}$Given:P=CV⁻¹  (1)Volume=V=F·t  (2)Substitute in 1P=C(F·t)⁻¹  (3)Divide (3) by FP/F=C/F ²(l/t)R=C/F ²(l/t)where t is the total time of inspiration/expiration.

Thus, in the laboratory environment flow pressure and time are measuredas a routine medical assessment in order to determine flow limitation.Generally, one needs to look at both pressure and flow. Pressure ismeasured by the catheter and then it is compared to the criteria byvisual inspection of a plateau of 1 cm of pressure and whether there isalso present a decrease in flow.

However the inventors herein have found a polynomial functionrelationship between flow and time using derivatives. This relationshipdoes not require that pressure be specifically measured, such as with acatheter, and therefore the novel method obviates the need for theinvasive catheter. As described herein, this new method was validated bythe inventors by comparing 440 breaths from multiple subjects to theresults of the conventional method. The novel method produced resultsthat were 98% correlated to the conventional method.

Software Implementation

The inventors herein have designed new codes that facilitate the rapidanalysis of large quantities of data. Generally speaking, the codeoperates to form a moving average in different arrays and special curvefit of the function to the polynomial, and transfers data from MicroSoftExcel®. The algorithm performs calculations of derivatives andidentifies the flow limitation by criteria. Unnecessary data is deleted.Moreover, the algorithm calculates the average of pressure flow fromdifferent breaths and draws all of them in stacked relation on onecomposite graph. Thus, there is formed a precise composite loop ofseveral breaths added on top of each other.

An illustrative source code for a macro (Macro 1) is as follows:   SubGroup_Normalize_Chart( )   Cells.Select   ActiveWindow.Zoom = 75  Selection.NumberFormat = “0.00”   Range(“A1:O25”).Select  Selection.Cut Destination:=Range(“P1:AD25”)   Range(“P1:AD25”).Select  ActiveWindow.LargeScroll ToRight:=−1   Range(“A1:O25”).Select  Selection.Delete Shift:=xlUp Dim BOTTOM As Long, COUNTER As Integer, XAs Long, MVAL As Single Dim STOPPER As Long   BOTTOM = Cells(16384,1).End(xlUp).Offset(−1, 0).Row   COUNTER = 1   Range(“A3”).Select   X =3   Do Until Cells(X, 1).Row > BOTTOM     Cells(X, 3).Value = Cells(X,2).Value − MVAL     Cells(X, 4).Value = Cells(X, 3).Value / Cells(X,1).Value     If Left(Cells(X − 1, 1).Value, 5) < > “Group” Then       IfCells(X − 1, 1).Value < −0.0001 And Cells(X, 1).Value >= −0.0001 Then        Cells(X, 1).Select         Selection.EntireRow.Insert        BOTTOM = BOTTOM + 1         ActiveCell.Value = “Group ” &COUNTER         If Abs(ActiveCell.Offset(−1, 0).Value) <Abs(ActiveCell.Offset(1,           0).Value) Then           MVAL =ActiveCell.Offset(−1, 1).Value         Else           MVAL =ActiveCell.Offset(1, 1).Value         End If         COUNTER = COUNTER +1       End If     End If     X = X + 1   Loop   InvertedPlot  Find_Flex_Point End Sub Private Sub InvertedPlot( ) Dim TOP As Long,BOTTOM As Long, MYLOC As String Dim MYTITLE As String, XPLOT As String,YPLOT As String   CURRENT = ActiveSheet.Name   Range(“A1”).Select  Cells.Find(What:=“Group 1”, After:=ActiveCell, LookIn:=xlFormulas,LookAt_(—)     :=xlPart, SearchOrder:=xlByRows, SearchDirection:=xlNext,MatchCase:= _(—)     False).Activate   Do UntilIsEmpty(ActiveCell.Value) Or ActiveCell.Value = “END DATA”     MYTITLE =ActiveCell.Value     ActiveCell.Offset(1, 1).Select     TOP =ActiveCell.Row     BOTTOM = ActiveCell.End(xlDown).Row    Range(Cells(TOP, 3).Address, Cells(BOTTOM,3).End(xlDown).Address).Select     MYLOC = Selection.Address    Charts.Add     ActiveChart.ChartType = xlXYScatterLines    ActiveChart.SetSourceData Source:=Sheets(CURRENT).Range(MYLOC), _(—)      PlotBy:=xlColumns     XPLOT = “=”‘ & CURRENT & ’“!R” & TOP &“C3:R” & BOTTOM & “C3”     YPLOT = “=”‘ & CURRENT & ’“!R” & TOP & “C1:R”& BOTTOM & “C1”     ActiveChart.SeriesCollection(1).XValues = XPLOT    ActiveChart.SeriesCollection(1).Values = YPLOT    ActiveChart.Location Where:=xlLocationAsNewSheet     WithActiveChart       .HasTitle = True       .ChartTitle.Characters.Text =MYTITLE       .Axes(xlCategory, xlPrimary).HasTitle = True      .Axes(xlCategory, xlPrimary).AxisTitle.Characters.Text =“Pressure”       .Axes(xlValue, xlPrimary).HasTitle = True      .Axes(xlValue, xlPrimary).AxisTitle.Characters.Text = “Flow”    End With     Sheets(CURRENT).Activate     Cells(BOTTOM + 1,1).Select   Loop End Sub Private Sub Find_Flex_Point( ) Dim X As Long, YAs Long, Z As Long, FIRST As Long, LAST As Long   ‘ Find the top of thefirst group   FIRST = Range(“C3”).End(xlDown).Offset(2, 0).Row   ‘ Setthe top of the table area   Z = FIRST − 1   Cells(Z, 7).Select   ‘ Writetable titles   ActiveCell.Offset(0, −1).Value = “Group”  ActiveCell.Value = “FLOW”   ActiveCell.Offset(0, 1).Value = “Adj.Millar”   ActiveCell.Offset(0, 2).Value = “Resistance”   Z = Z + 1   ‘Loop through and find the flex point in each group   Do UntilIsEmpty(Cells(FIRST, 3).Value)     ‘ find the bottom row of the currentgroup     LAST = Cells(FIRST, 3).End(xlDown).Row     ‘ loop through thedata to find the first value > 0.2.     For X = FIRST To LAST       IfCells(X, 1).Value >= 0.2 Then         ‘ preserve the row number        Y = X         Exit For       End If       X = X + 1     Next X    ‘ determine if the prior value is closer than the current value    If Abs(Cells(Y, 1).Value − 0.2) > Abs(Cells(Y − 1, 1).Value − 0.2)Then Y = Y − 1     ‘ Calculate and write the desired value in column D    ‘ Cells(Y, 4).Value = Cells(Y, 3).Value / Cells(Y, 1).Value     ‘Fill the table with the Group and values     Cells(Z, 6).Value =Cells(FIRST − 1, 1).Value     Cells(Z, 7).Value = Cells(Y, 1).Value    Cells(Z, 8).Value = Cells(Y, 3).Value     Cells(Z, 9).Value =Cells(Y, 4).Value     ‘ Increment the counters to the proper location    FIRST = LAST + 2     Z = Z + 1   Loop  ActiveCell.End(xlDown).Offset(0, 2).Select End Sub

This macro receives raw data corresponding, inter alia, to flow and timeas the data is delivered to the polygraph (not shown). The data then isexported, in this specific illustrative embodiment of the invention, toa MicroSoft Excel® spread sheet (not shown). At this point, the macro isready to do its action. The data in the spread sheet then is dividedinto breaths based on the fact that each breath would start with Flow=0and that inspiration precedes expiration. Then, adjusted pressure wouldbe added as a new column to make sure that the first coordinate of everybreath would be (Pressure=0, Flow=0). This is normalizing the data,including time. Next, a fourth column is added as the resistance, whereresistance=adjusted Pressure/Flow. Then, a table is created presentingthe value of resistance at fixed flow (flow=0.20 L/s) for every breath.Finally, an X-Y graph is plotted for every breath where adjusted time inthe x-axis and flow in the y-axis.

There is provided a second macro (Macro 2) that performs the same stepsmention in the relation to Macro 1 up to calculating the adjustedpressure step. Then the adjusted time as x-axis and flow y-axis isconsidered. Next, a curve fitting of the inspiratory rising limbflow-time, a mathematical polynomial function F(P)=A t³+B t²+Ct+D. whereA, B, C, and D are the coefficients (constants). The software willcalculate the coefficients (A, B, C, D), and calculate the derivative ofthe mathematical model, which represents the slope. If the derivative atthe maximum actual flow is zero or negative then the breath, aspreviously discussed, is inspiratory flow limited

Although the invention has been described in terms of specificembodiments and applications, persons skilled in the art may, in lightof this teaching, generate additional embodiments without exceeding thescope or departing from the spirit of the claimed invention.Accordingly, it is to be understood that the drawing and description inthis disclosure are proffered to facilitate comprehension of theinvention, and should not be construed to limit the scope thereof.

1. A method of measuring upper airway resistance of a human patient, themethod comprising the steps of: obtaining air pressure data from an airpressure data signal corresponding to a plurality of breathing cycleswhile the human patient is asleep; obtaining air flow data from an airflow data signal corresponding to the plurality of breathing cycleswhile the human patient is asleep; transferring the air pressure dataand the air flow data to a processor; storing the air pressure data andthe air flow data in respective correlated storage regions of a matrixprogram system of the processor; segregating the air pressure data andthe air flow data in the matrix program of the processor intocorresponding breathing cycles of the human patient; computingnormalized air pressure data to achieve a predetermined normalized airpressure value to correspond with a predetermined point for eachbreathing cycle of the human patient; producing a correlation of the airflow data against normalized air pressure data; curve-fitting onto thecorrelation of the air flow data against normalized air pressure data acurve corresponding to a predetermined multiple term mathematicalfunction; computing the value of the coefficients of the predeterminedmultiple term mathematical function; and computing the derivative of thepredetermined multiple term mathematical function.
 2. The method ofclaim 1, wherein in said step of curve-fitting onto the correlation ofthe air flow data against normalized air pressure data a curve, thepredetermined multiple term mathematical function is a quadraticfunction, F(P)=AP²+BP+C, where A, B, and C are coefficients.
 3. Themethod of claim 1, wherein in said step of curve-fitting onto thecorrelation of the air flow data against normalized air pressure data acurve, the predetermined multiple term mathematical function is a threeterm polynomial function F(P)=AP³+BP²+CP+D, where A, B, C, and D arecoefficients.
 4. The method of claim 3, wherein said step of computingthe derivative of the three term polynomial function corresponds to therelationship:$\frac{\mathbb{d}F}{\mathbb{d}P} = {{3{AP}^{2}} + {2{BP}} + {C.}}$ 5.The method of claim 4, wherein there is provided the step of determiningthat a breath is inspiratory in response to the derivative of the threeterm polynomial function having a value of zero or positive, whereby${\frac{\mathbb{d}F}{\mathbb{d}P} \geq 0}->{{IFL}.}$
 6. The method ofclaim 4, wherein there is provided the step of determining that a breathis inspiratory in response to the derivative of the three termpolynomial function having a negative value, whereby${\frac{\mathbb{d}F}{\mathbb{d}P} < 0}->{{NIFL}.}$
 7. The method ofclaim 1, wherein there is further provided the step of computing aresistance corresponding to the reciprocal of coefficient C, wherebyResistance=1/C.
 8. A method of determining a flow-limitingcharacteristic of the upper airway of a human patient, the methodcomprising the steps of: obtaining air pressure data from an airpressure data signal corresponding to a plurality of breathing cycleswhile the human patient is asleep; obtaining air flow data from an airflow data signal corresponding to the plurality of breathing cycleswhile the human patient is asleep; transferring the air pressure dataand the air flow data to a processor; storing the air pressure data andthe air flow data in respective correlated storage regions of a matrixprogram system of the processor; segregating the air pressure data andthe air flow data in the matrix program of the processor intocorresponding breathing cycles of the human patient; computingnormalized air pressure data to achieve a predetermined normalized airpressure value to correspond with a predetermined point for eachbreathing cycle of the human patient; and computing the flow-limitingcharacteristic of the upper airway of a human patient as a function ofnormalized air pressure data divided by corresponding air flow data. 9.The method of claim 8, wherein the matrix program system is aspreadsheet program system, the air pressure data and the air flow databeing arranged in respective spreadsheet columns correlated by rows. 10.The method of claim 9, wherein said step of computing normalized airpressure data comprises the further step of storing the normalized airpressure data in a respective spreadsheet column correlated by rows intocorresponding breathing cycles of the human patient.
 11. The method ofclaim 8, wherein each breathing cycle of the human patient is determinedin relation to the predetermined point thereof corresponding to thepredetermined normalized air pressure value.
 12. The method of claim 11,wherein there is further provided the step of computing theflow-limiting characteristic of the upper airway of a human patient foreach of the plurality of breathing cycles.
 13. The method of claim 11,wherein the predetermined normalized air pressure value corresponds to azero value.
 14. The method of claim 13, wherein each breathing cycle ofthe human patient is further determined in relation to the predeterminedpoint thereof corresponding to the air flow data having a zero value.15. The method of claim 8, wherein the air pressure data and the airflow data are sampled a plurality of times during each breathing cycle.16. The method of claim 15, wherein said step of computing theflow-limiting characteristic of the upper airway of a human patient isperformed a corresponding plurality of times during each breathingcycle.
 17. The method of claim 16, wherein said step of computing theflow-limiting characteristic of the upper airway of a human patient isperformed a corresponding plurality of times during each breathing cycleand during which the air flow data has a predetermined value.
 18. Themethod of claim 17, wherein there is further provided the step ofcorrelating the air flow data and the normalized pressure data to form adata correlation in a data correlation array, and the predeterminedvalue of the air flow data is determined within a substantially linearportion of the data correlation.
 19. The method of claim 18, wherein thepredetermined value of the air flow data is approximately between 0.00L/s and 0.22 L/s.
 20. The method of claim 18, wherein the predeterminedvalue of the air flow data is approximately 0.20 L/s.
 21. The method ofclaim 18, wherein there is provided the further step of computing aslope of the correlated air flow data and normalized pressure datawithin the substantially linear portion of the data correlation.
 22. Themethod of claim 17, wherein there is provided the further step ofproducing a data array corresponding to the flow-limiting characteristicwherein the normalized air pressure data corresponds to the x-axis andthe air flow data corresponds to the y-axis.
 23. A method of measuringupper airway resistance of a human patient, the method comprising thesteps of: obtaining air pressure data from an air pressure data signalcorresponding to a plurality of breathing cycles while the human patientis asleep; obtaining air flow data from an air flow data signalcorresponding to the plurality of breathing cycles while the humanpatient is asleep; transferring the air pressure data and the air flowdata to a processor; storing the air pressure data and the air flow datain respective correlated storage regions of a matrix program system ofthe processor; segregating the air pressure data and the air flow datain the matrix program of the processor into corresponding breathingcycles of the human patient; computing normalized air pressure data toachieve a predetermined normalized air pressure value to correspond witha predetermined point for each breathing cycle of the human patient;producing a correlation of the air flow data against normalized airpressure data; curve-fitting onto the correlation of the air flow dataagainst normalized air pressure data a curve corresponding to a threeterm polynomial function F(P)=AP³+BP²+CP+D, where A, B, C, and D arecoefficients; and computing the value of the upper airway resistance ofa human patient as an inverse function of coefficient C of the threeterm mathematical function, C, wherebyResistance=1/C.
 24. The method of claim 23, wherein each breathing cycleis defined at an onset point where inspiratory flow is zero.
 25. Themethod of claim 23, wherein each breathing cycle is defined at an onsetpoint where supraglottic pressure has been normalized to zero.
 26. Themethod of claim 23, wherein there is further provided the step ofcomputing the derivative of the three term polynomial function inaccordance with the relationship:$\frac{\mathbb{d}F}{\mathbb{d}P} = {{3{AP}^{2}} + {2{BP}} + {C.}}$ 27.The method of claim 26, wherein there is further provided the step ofdetermining the presence of inspiratory flow limitation in response tothe derivative of the three term polynomial function.
 28. The method ofclaim 23, wherein said step of obtaining air flow data from an air flowdata signal comprises the further step of recording the air flow datasignal on a polygraph.
 29. The method of claim 28, wherein said step ofstoring the air pressure data and the air flow data in respectivecorrelated storage regions of a matrix program system of the processorcomprises the step of exporting the air pressure data and the air flowdata to a first Excel® spreadsheet.
 30. The method of claim 29, whereinthere is further provided the step of plotting a graphicalrepresentation of adjusted time along the x-axis and flow along they-axis.
 31. The method of claim 30, wherein there is further providedthe step of curve fitting an inspiratory rising limb flow-time curve toa mathematical polynomial function F(P)=A t³+B t²+Ct+D. where A, B, C,and D are coefficients.
 32. The method of claim 31, wherein there isfurther provided the step of calculating a derivative of themathematical polynomial function.
 33. The method of claim 32, whereinthere is further provided the step of exporting the value of thederivative of the mathematical polynomial function to a second Excel®spreadsheet.
 34. The method of claim 33, wherein there is furtherprovided the step of determining whether a breath is or is not flowlimited, in response to the value of the derivative of the mathematicalpolynomial function.